Filling driveway concrete gap

I just bought this house and am trying to set things right. There is a 1-1/2" gap between the driveway and the garage slab. (I have a similar gap at the other side of the house between the patio and foundation.)

I would like to keep water from filling this area, to reduce heaving in the winter. What would be the proper materials to use? Some kind of urethane rope stuffer, then something on top?